Density of a thin film billiard reflection pseudogroup in a Hamiltonian symplectomorphism pseudogroup

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to canonical symplectic form. We consider an arbitrary C?-smooth hypersurface ? ? ?n+1 that is either a global strictly convex closed hypersurface, or germ hypersurface. deal pseudogroup generated compositional ratios reflections and its small deformations. In case when we show C?-closure latter contains Hamiltonian diffeomorphisms between domains in phase cylinder: intersecting transversally. prove analogous local result germ. The derivatives above differences deformation parameter are vector fields calculated Ron Perline. To main results, find Lie algebra them C?-density fields. also analogues results for Riemannian manifolds.